Mrs. Coolidge's response was to indicate that she had in fact arranged four performances of the quartet - one was at the Lennox Gallery, New York Public Library, and another at her Founder's Day Concert at the Library of Congress, Washington (on 30 October). This
was not quite Bridge's point, of course, but it was a cogent reminder
of her continued support. Another reminder took place at the Founder's Day Concert when Mrs Coolidge awarded Bridge her Foundation Medal, a fitting tribute to a lifetime's devotion to chamber music and a close friendship of over fifteen years. Bridge was deeply touched.

Four years earlier, during the summer of 1934, Bridge had wanted to present Mrs Coolidge with a birthday composition for her 1935 Founder's Day Concert, and had begun the Divertimenti for wind quartet. The germ of the idea to compose this piece had been planted by
Georges Barrère, the French flautist, whom Bridge had met in Chicago during his third American visit, and whose friendship he renewed during his two subsequent visits. However, Bridge refrained from mentioning his plan to Mrs Coolidge until early in 1938:

I am putting together some unfinished things for Flute Oboe Clarinet and Bassoon which I always intended, at some time or other to have up my sleeve for you at a birthday concert at Washington, and, in my usual stupid way, somehow they wouldn't tie themselves into a proper bundle, but I think they will soon.34

He had composed two movements, during June and July 1934, but only returned to them three years later. He revised them and wrote another two, the completion dates of the first three movements coinciding almost exactly with those for the Fourth String Quartet's three movements. Following a private play-through that Bridge arranged in late May, 1938, he sent a score and parts to Mrs Coolidge, suggesting that she could have the work performed at her Founder's Day Concert that year. However, his hopes were dashed. She pointed out that, in consequence of contractual agreements, only the Coolidge Quartet could perform at these concerts, and, if he wanted the Divertimenti played, the only solution would be for Bridge to arrange them for string quartet.35 Bridge had to accept this with good grace: